Two common examples of dependent types are dependent functions and dependent pairs. A dependent function's return type may depend on the value (not just type) of an argument. A function that takes a positive integer "n" may return an array of length "n". (Note that this is different from polymorphism where the type is an argument.) A dependent pair may have a second value that depends on the first. It can be used to encode a pair of integers where the second one is greater than the first.

Dependent types add complexity to a type system. Deciding the equality of dependent types in a program may require computations. If arbitrary values are allowed in dependent types, then deciding type equality may involve deciding whether two arbitrary programs produce the same result; hence type checking may become undecidable.

Predicate logic is an extension of propositional logic, adding quantifiers. Howard and de Bruijn extended lambda calculus to match this more powerful logic by creating types for dependent functions, which correspond to "for all", and dependent pairs, which correspond to "there exists".[2]

Dependent types can be, very loosely speaking, imagined to be similar to the type of an indexed family of sets. More formally, given a type in a universe of types , one may have a family of types which assigns to each term a type . A function whose codomain varies depending on its argument is a dependent function, and the type of this function is called a dependent type, dependent product type or pi-type. For this example, the dependent type would be written as

or as

If B is a constant, the dependent type becomes an ordinary function . That is, is judgementally equal to the function type.

For example, writing for -tuples of real numbers, then would be the type of functions which, given a natural numbern, return a tuple of real numbers of size n. The usual function space arises as a special case when the range type does not actually depend on the input, e.g. is the type of functions from natural numbers to the real numbers, which is written as in the simply typed lambda calculus.

Polymorphic functions are an important example of dependent functions, that is, functions having dependent type. Given a type, these functions act on elements of that type (or on elements of a type constructed (derived, inherited) from that type). A polymorphic function returning elements of type C would have a polymorphic type written as

Let be sigma-type quantifying over type with predicate . There exists an such that holds if and only if is inhabited. For example, a is less than b iff there exists a natural number n and a proof that a+n=b.

Henk Barendregt developed the lambda cube as a means of classifying type systems along three axes. The eight corners of the resulting cube-shaped diagram each correspond to a type system, with simply typed lambda calculus in the least expressive corner, and calculus of constructions in the most expressive. The three axes of the cube correspond to three different augmentations of the simply typed lambda calculus: the addition of dependent types, the addition of polymorphism, and the addition of higher kinded type constructors (functions from types to types, for example). The lambda cube is generalized further by pure type systems.

The system of pure first order dependent types, corresponding to the logical framework LF, is obtained by generalising the function space type of the simply typed lambda calculus to the dependent product type.

The system of second order dependent types is obtained from by allowing quantification over type constructors. In this theory the dependent product operator subsumes both the operator of simply typed lambda calculus and the binder of System F.

The Curry–Howard correspondence implies that types can be constructed that express arbitrarily complex mathematical properties. If the user can supply a constructive proof that a type is inhabited (i.e., that a value of that type exists) then a compiler can check the proof and convert it into executable computer code that computes the value by carrying out the construction. The proof checking feature makes dependently typed languages closely related to proof assistants. The code-generation aspect provides a powerful approach to formal program verification and proof-carrying code, since the code is derived directly from a mechanically verified mathematical proof.